TECHNICAL NOTES

Elastic settlement of arbitrarily shaped foundations embedded in half-space

G. GAZETAS,* J. L. TASSOULAS,t R. DOBRY* and M. J. OROURKE*

The settlement of foundations under working loads which induce relatively small strains in the soil is frequently estimated in practice on the basis of results from the theory of elasticity. Despite its shortcomings in modelling actual soil behaviour, elastic theory may be quite useful especially in predicting immediate settlements on saturated clayey deposits, provided that an appropriate value of undrained secant Youngs modulus has been experimentally determined.

Most of the available elastic solutions refer to loads acting directly on the ground surface. To estimate the settlement of an embedded founda- tion without resorting to expensive numerical (mainly finite element and boundary element) techniques, engineers apply reduction factors to the settlement of the corresponding surface foundation. While most available reduction fac- tors are purely empirical in nature, an approxi- mate solution by Fox (1948) has been particu- larly popular for embedded flexible rectangular foundations. Foxs solution has been reproduced in the form of a simple chart by Janbu, Bjerrum & Kjaernsli (1956). This chart is still widely used in geotechnical engineering, being recom- mended in several textbooks of soil mechanics and foundation engineering. Unfortunately, as Burland (1970) and Christian & Carrier (1978) have shown, Foxs factors may grossly exagger- ate the effect of embedment, when compared with the results of finite element analyses. This is hardly surprising: Foxs results, based on Mindlins solution for a point load within a half-space, refer to a uniformly loaded area surrounded by (and bonded to) an elastic con- tinuum, rather than to an embedded foundation. That is, Foxs results implicitly assume that part of the carried load is transmitted to the ground through tension between the upper side of the foundation mat and the overlying soil. However, no (net) tension can usually develop between soil and foundation and, in most cases, the overlying soil has been removed by excavation.

The results of the studies by Burland and

Discussion on this Technical Note closes on 1 January 1986. For further details see inside back cover. * Rensselaer Polytechnic Institute, Troy, New York. t The University of Texas, Austin.

Christian and Carrier are limited to uniformly loaded areas placed at the bottom of an open trench. Rigid foundations were not considered and the effects of contact between the soil and the foundation sidewalls were neglected. How- ever, Kaldjian (1969), Johnson, Christian0 & Epstein (1975) and Kausel & Ushijima (1979) have presented some results for circular and strip foundations with vertical sides in perfect contact with the soil.

In this Note a simple realistic analytical ex- pression is developed for estimating the vertical elastic settlement of arbitrarily shaped rigid foundations embedded in a reasonably uniform and deep soil deposit, modelled as a homogene- ous half-space (Fig. 1). The expression is applic- able to a large range of embedment depths and a variety of solid base shapes, ranging from circular to strip and including rectangles of any aspect ratio as well as odd shapes differing sub- stantially from rectangular or circular. (Annular base shapes are excluded, however.) Further, the expression encompasses the wide variation in the type of contact between foundation sidewalls and surrounding soil, .between the ex- treme cases of complete perfect contact and of no contact at all.

The development of the proposed expression was based on an improved qualitative under- standing of the effects of embedment, substan- tiated quantitatively by the results of extensive rigorous parametric studies using the boundary element method, and including numerous analytical and numerical results available in the literature. It is emphasized that the proposed algebraic expressions are curve fits of the analyt- ical data; hence, the accuracy of the settlement computed from the proposed expression may be not better than lo-20%. Note, however, that discrepancies of about 10% or more have also been observed among the results obtained from different rigorous solutions to the same problem. Such discrepancies arise because of the different assumptions regarding the behaviour of the soil- foundation interface (rough versus relaxed boundary conditions), different solution methods (such as integral transform techniques, semi- analytical procedures, finite elements and the boundary element method) and different de-

339

340 TECHNICAL NOTES

Rigid foundy Section t-1

Fig. 1. Problem geometry

grees of precision with which the calculations have been carried out.

SETTLEMENT OF SURFACE FOUNDATIONS

The elastic settlement of a rigid foundation of arbitrary shape resting on the surface of a homogeneous half-space of modulus E and Poissons ratio Y, and carrying a total vertical force P may be estimated from

(la)

in which -0.38

(lb)

where Ab is the plan area of the foundation-soil contact surface, and 2L and 2B are the length and width of the rectangle circumscribed to the actual contact surface, as illustrated in Figs 1 and 2(b). The dimensionless shape parameter

A,,/4L2 is equal to unity for a square, 0.785 for a circle and zero for an infinitely long strip. For rectangles Ab/4L2 = B/L, i.e. it reduces to the inverse of the aspect ratio.

Expression (lb) has been derived by fitting analytical and numerical results from the numer- ous publications listed in Table 1, including sev- eral results by the Authors. As shown in Fig. 2(a), the scatter of the results around the curve of equation (lb) is very small for all the cases considered. For an arbitrary solid shape (whose actual area A,, is not less than about 40% of the area 4BL of the circumscribed rectangle) the true elastic surface settlement is expected to be within 5-10% of the value predicted by equa- tion (1).

EFFECTS OF EMBEDMENT ON FOUNDATION SETTLEMENT

There are three possible effects of embedment on the vertical settlement of a rigid foundation.

TECHNICAL NCYIES 341

P shape =0.45 (Ab/4L2)-0,3e

Circle :- -1 28 a I__ _:

2L

L/B = 1

Ab/(4L2P0.7B5

L/B=2

A,/i4L2 PO.39

Rhombus

L/8=2.3 L/B=2.4

A,/(4?1= 0.22 A,/t4L2,=0.21

Ellipse

L/B=4 L/B=3

A,/(4L2 )O 20 A,/14L2)=0.25

2L

I I I I I 0.2 0,4 0.6 0.0 1

A,

4L2

(a) (b)

Fig. 2. (a) Surface settlement shape factor versus foundation shape parameter (the shapes of the data points correspond approximately to those of Table 1) and (b) examples of non-rectangular base shapes and correspond- ing ci;eums&bed rectangles

First of all, for the usual situation in which soil stiffness increases with depth, embedded found- ations transmit load to stiffer soil beneath them than do surface foundations. Hence, other things being equal, smaller settlement is expected for an embedded foundation, although settlements may be the same or even increase compared with surface foundations if the soil stiffness de- creases with depth. This important effect of em- bedment will not be further addressed herein. However, when applying the proposed method to a practical situation

(a) it must be ensured that the soil deposit is indeed reasonably homogeneous and deep

(b) a representative value of soil modulus for the particular depth of embedment must be established.

The other two effects which modify the be-

haviour of embedded foundations and are addres- sed in this Note are referred to as the trench effect and the sidewall contact effect. Both of these effects are illustrated schematically in Fig. 3.

The trench effect refers to the fact that, even in a perfectly homogeneous half-space, the set- tlement of a foundation placed at the bottom of an open trench is smaller than that of the same foundation placed on the ground surface. To understand why, visualize a horizontal plane surface passing through the foundation base. For the surface foundation this plane deforms free of any external stress. However, for the embedded foundation, normal and shear stresses from the overlying soil restrict the movement of that hori- zontal plane (Fig. 3(a)), thereby reducing the foundation settlement from pSUr to pvench = ptrenchpsur. There is analytical as well as experi-

342 TECHNICAL NOTES

Table 1. Surface foundations: data of Fig. 2(a) and their sources*

Shape LIB

Square 1 1 Circle 1 0.79 Hexagon (regular) 1 0.65 Rectangle 2 0.5 Triangle (equilateral) 1.15 0.44 Ellipse 2 0.39 Semicircle 2 0.39 Ellipse 3 0.26 Triangle (45-45-90) 2 0.25 Rhombus (60) 2 0.25 Rectangle 4 0.25 Triangle (30-60-90) 2.3 0.25 Rhombus (45) 2.4 0.22 Ellipse 4 0.21 Rhombus (30) 3.1 0.20 Ellipse 6 0.14 Rectangle 8 0.13 Rectangle 10 0.10 Rectangle 20 0.05 Strip 30 0

L

* References: Gorbunov-Possadov & Serebra- janyi (1961); Dominguez & Roesset (1978); Selvadurai (1979); Barkan (1962); Borodachev (1964); Sneddon (1951); Conway & Farnham (1968); Savidis (1977); Riicker (1982); Mus- khelishvili (1953); Poulos & Davis (1974); Au- thors.

mental evidence supporting the existence of the trench effect. Erdens (1974) experimental work on dry sand revealed that even with no side contact along the embedment depth, the effect

trench effect

(b) Fig. 3. Embedded foundation: schematic illustration of trench and sidewall contact effects

of embedment [to reduce the settlement] is not caused solely by an increase in shear modulus beneath the footing. Moreover, Erden (1974) showed that the settlement of a foundation in an open trench is essentially the same as, and in fact even somewhat smaller than, that of a sur- face foundation when the free ground surface is pneumatically loaded with a normal pressure equivalent to the overburden stress at basemat level. Apparently, both normal and shear over- burden stresses, as sketched in Fig. 3(a), play a role in reducing the trench settlement. The num- erical data needed to develop a relationship for p trench = Ptrencll PSW I have been compiled from several publications (listed in Table 2) dealing primarily with circular and strip foundations, and from a comprehensive parametric study by the Authors for embedded circular, strip and rectangular foundations with aspect ratios of up to 6, and for two values of Poissons ratio: 0.33 (typical of cohesionless soil) and 0.49 (typical of saturated clays). Fig. 4 presents this information in the form of a graph of *trench as a function of the dimensionless parameter (D/B)[ 1 +$x (A,/4L2)]. Determined by trial and error so that the results plot within a relatively narrow band, this parameter combines the relative depth of embedment and the base shape. An approxi- mate expression for the trench effect coefficient I*,_~,, was developed by passing a straight line through the analytical data points

f.LLtrech= 1-0.04~(1+5$) (2)

Combinedtrenchzond sidewall contact effects !-!

TECHNICAL NOTES 343

Table 2. Embedded foundations: data of Figs 4 and 5, and their soorces

Symbol Y Shape* Reference

0.25

0.25

I 0.49

0.33

0.33

0.40

0.33

0.331 0.49J

0.25 0.33 1 0.49

Circle

Circle (uniform applied pressure)

Square

Circle (partial sidewall contact)

Circle

Rectangle L/B = 4

Kaldjian (1969)

Burland (1970), Christian & Carrier (1978)

Dominguez & Roesset (1978)

Tassoulas (1981)

Authors

Authors

Rectangle L/B = 6 Authors

strip Johnson et al. (1975); Authors

* Unless otherwise stated, results are for a rigid base and a rigid sidewall.

The error of equation (2) is in most cases less applied load is transmitted to the ground than 10%. through shear tractions acting along the vertical

The third phenomenon to be considered is the sides of the wall (Fig. 3(b)). This additional load sidewall contact effect. When the vertical path leads to a further reduction in the settle- sidewalls of an embedded foundation are in ment of an embedded foundation with sidewall contact with the surrounding soil, part of the contact, compared with that in an open trench.

( ,+4Ab 3 4L2 >

o.6\ 0 2 4 6

Fig. 4. Trench settlement reduction factor versus shape-dependent normafized depth parameter (for references see Table 2)

344 TECHNICAL NOTES

. .e 0.6 I ?

0 2 4

A, Ab

Fig. 5. Sidewall contact settlement reduction factor versus wall-base area ratio (for references see Table 2)

This reduced settlement pernb is evaluated from the open trench settlement ptrench through the sidewall coefficient F.,,,,

The numerical data for developing an expression for F,,,~, were obtained by dividing the respec- tive settlements of an embedded foundation with complete sidewall-soil contact ((J_,,~) and with absolutely no contact (ptrench). The sources of these data are also included in Table 2.

Figure 5 shows the sidewall coefficient pwall as a function of the effective wall-base area ratio AJAb, where A, is the area of the sidewall-soil interface and As is the area of the basemat. The choice of the parameter A,/A, is quite natural, since effectively the coefficient (L,,,, weighs the contributions to stiffness from the sides and from the base. Indeed, the scatter of the data points in Fig. 5 about the mean curve

P,,,~,,= l-0.16 $ ( )

O-54 (4)

b

is fairly small and essentially independent of the basemat shape and of the relative depth.

In conclusion, the settlement of an embedded foundation may be approximated by

Per& = Psur(LtrenchCLwall

Substituting equations (l), (2) and (4)

(5)

Pemb = 0.45 & (I - v) ($)-(=

X[l-0.04;(1+;$)]

x [I-O.16 (2)] (6)

is obtained for the settlement of a foundation embedded in a homogeneous half-space and having an arbitrary but solid base shape. For cases where there is doubt about the quality of contact between the sidewall and the surround- ing soil, the engineer may want to apply a reduction factor q (between zero and unity) to the wall-base area ratio, i.e. to substitute qA,/A, for A,/A,, in equation (6).

ILLUSTRATIVE EXAMPLE Equation (6) is now directly applied to esti-

mate the settlement of the embedded founda- tion portrayed in Fig. 6. In plan, the continuous and rigid foundation basemat is composed of two rectangles, one square and one semicircle. Part of the basemat perimeter is connected to a vertical sidewall 7.5 m high which is in contact with the surrounding soil. The sidewall-soil con- tact quality factor q is assumed to be 0.75. No sidewall (and hence no contact) exists along defgh, because of a hypothetical open space next to the foundation structure. The geometric parameters of the foundation and the elastic soil parameters are shown in Fig. 6. Of interest is

TECHNICAL NOTES

plan

I no contact

kc + 45+; 14.0 -4

(open space)

section aa I

deep deposit of stiff clay:

E=6.0 MPa,1/=0.35

Foundation geometric parameters:

2L=27,5m,20=105m Ab19924m

L/B = 2.62, Ab/(4L2) = 0 263

#=7.5m, D/B = 1.43, A,_,= 57.5 x7.5 = 431 3m2

AwAb z 2.16, estlmoted q E 0.75, q A,/Ab1.62

Fig. 6. Geometry and material parameters of aa illustrative example

the settlement due to a central vertical force P = 8 MN. Ignoring as secondary the incidental rotation that may occur from the lack of com- plete symmetry of the soil reactions at the base and the walls, with respect to the centroid C of the base, equation (6) gives

8(1-0.3P) P= 0.45 x 0.263~-78 6x 13.75

x

x(1-0.16x l.62S4)

= 0.085 x 0.748 x 0.923 x 0.792

=O.O46m=46mm

CONCLUSION A versatile simple analytical expression (equa-

tion (6)) is proposed for estimating the vertical elastic settlement of foundations with any solid basemat shape and embedded in reasonably deep and uniform soil deposit. The expression is applicable for a constant depth of embedment but encompasses all possible types of contact

346 TECHNICAL NOTES

between sidewalls and surrounding soil: com- plete and perfect contact, partial contact and no contact at all.

It may be noted that, although the rigorous numerical and analytical results on which equa- tion (6) is based were obtained for a homogene- ous half-space, the same conceptual framework can be readily extended to the more realistic case of a soil profile consisting of a half-space below the base and of an overlying side soil (backfill) of a different material. Also, although most of the basic data relate to a perfectly rigid basemat, equation (6) may provide sufficiently accurate estimates of the average settlement of flexible foundations.

REFERENCES

Barkan, D. D. (1962). Dynamics of bases and founda- tions. Russian translation. New York: McGraw- Hill.

Borodachev, N. M. (1964). Determination of the set- tlement of rigid plates. Soil Mech. Fdn Engng (USSR) 1, 210-214.

Burland, J. B. (1970). Proc. Conf. Insitu Investigations in Soils and Rocks. Discussion on session A, pp. 61-62. London: British Geotechnical Society.

Christian, J. T. & Carrier, W. D. (1978). Janbu, Bjerrum and Kjaernslis chart reinterpreted. Can. Geotech. J. 15, 123-128.

Conway, H. D. & Farnham, K. A. (1968). The rela- tionship between load and penetration for a rigid flat-ended punch of arbitrary cross-section. Inr. J. Engng Sci. 6, 489-496.

Dominguez, J. & Roesset, J. M. (1978). Dynamic stiffness of rectangular foundations. Research Re- port R78-20, Massachusetts Institute of Technology.

Erden, S. M. (1974). Influence of shape and embed- ment on dynamic foundation response. PhD thesis, University of Massachusetts.

Fox, E. N. (1948). The mean elastic settlement of a uniformly loaded area at a depth below the ground surface. Proc. 2nd Int. Conf. Soil Mech. Fdn Engng, Rotterdam, 1, 129-132.

Gorbunov-Possadov, M. I. & Serebrajanyi, V. (1961). Design of structures upon elastic foundations. Proc. 5th Int. Conf. Soil Mech. Fdn Engng, Paris 1, 643-648.

Janbu, N., Bjerrum L. & Kjaernsli, B. (1956). Vei- ledning ved losning av fundamentering soppgaver. Nonv. Geotech. Inst. Publ. 16, 30-32.

Johnson, G. R., Christiano, P. & Epstein, H. I. (1975). Stiffness coefficients for embedded footings. J. Geotech. Engng Diu. Am. Sot. Civ. Engrs 101, GTR. 789-800.

Kaldjian, M. J. (1969). Design procedures for dynami- cally loaded foundations. Discussion. J. Soil Mech. Fdns Div. Am. Sot. Civ. Engrs 95, SMl, 364-366.

Kausel, E. & Ushijima, R. (1979). Vertical and ror- sional stiffness of cylindrical footings. Research Re- port R79-6, Massachusetts Institute of Technology.

Muskhelishvili, N. I. (1953). Some basic problems of the mathematical theory of elasticity. Groningen: Noordhoff.

Poulos, H. G. & Davis E. H. (1974). Elastic solutions for soil and rock mechanics. New York: Wiley.

Riicker, W. (1982). Dynamic behavior of rigid founda- tions of arbitrary shape on a half space. Earrhq. Engng Stnccr. Dynam. 10, 675-690.

Savidis, S. A. (1977). Analvtical methods for the computation of wavefields. In Dynamic methods in soil and rock mechanics, vol. 1, p. 225. Rotterdam: Balkema.

Selvadurai, A. P. S. (1979). Elastic analyses of soil- foundation interaction. Amsterdam: Elsevier.

Sneddon. I. N. (1951). Fourier @ansforms. New York: McGraw-Hill.

Tassoulas, J. L. (1981). Elements for the numerical analysis of wave motion in layered media. Research Report RBl-2, Massachusetts Institute of Technology.